In a large covered pot, bring potatoes to a boil in enough salted water to cover by at least 1 inch. Lower heat to a simmer, and cook about 25-30 minutes or until potatoes are tender when pierced with a fork, but still firm. (Cooking time will vary according to the size and freshness of the potatoes).
Drain potatoes in colander, and briefly run cold water over the potatoes. Transfer drained potatoes back into the empty covered pot and set on a very low flame for just a minute or two, being careful not to scorch them. This will help to dry the potatoes and to ensure that they are not wet or soggy. Return potatoes to colander, and let them cool to the touch.
Peel and then slice the potatoes into one inch pieces.
In a large bowl, stir together the onions, apples, celery, dill pickles, sweet relish, and one tablespoon of the chopped parsley.
In a separate bowl, whisk together the sugar, vinegar, pickle juice, olive oil, mustard, mayonnaise, and the salt and pepper. Pour over the onion mixture, stirring well to coat evenly.
Add the potatoes and 3 chopped hardboiled eggs. Fold and stir gently to combine. Transfer to a wide and shallow serving dish.
Garnish with 2 quartered hardboiled eggs, tomato slices, and the torn lettuce. Sprinkle with remaining parsley and the paprika.